A general multi-valued hybrid fixed point theorem and perturbed differential inclusions

Abstract

Combining three basic multi-valued versions of Banach, Schauder and Tarski fixed point theorems, a general hybrid fixed point theorem for multi-valued mappings in Banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed Lipschitz, compactness and monotone conditions.

Introduction

Fixed point theory can be classified into three main streams, viz., algebraic, topological and geometrical fixed point theory. This classification is not rigid, but based on the nature of the major hypotheses involved in the fixed point theorems. If a fixed point theorem involves the mixed hypotheses of algebra, topology and geometry, then it is called a hybrid fixed point theorem and these hybrid fixed point theorems constitute a new stream of hybrid fixed point theory in the subject of nonlinear functional analysis. The hybrid fixed point theorems find numerous applications to other areas of mathematics such as differential and integral inclusions and equations. To the best of our knowledge, the first hybrid fixed point theorem for single-valued mappings is established by the Russian mathematician Krasnoselskii [31] which is also named in his name as Krasnoselskii fixed point theorem since long time. This fixed point theorem combines the metric fixed point theorem of Banach with a topological fixed point theorem of Schauder in a Banach space and Krasnoselskii himself applied it to some nonlinear integral equations of mixed type for proving the existence results under mixed Lipschitz and compactness conditions. Later in 1988, the present author proved a similar type of hybrid fixed point theorem in Banach algebras involving the product of two operators. It is worthwhile to mention that this hybrid fixed point theorem also has numerous applications to nonlinear differential and integral equations in Banach algebras. There are several extensions and generalizations of the above two hybrid fixed point theorems in the literature with some nice applications. Recently the multi-valued analogues of above two hybrid fixed point theorems have been obtained in Petruşel [33] and Dhage [10] and discussed their applications in the theory of differential and integral inclusions. There are other hybrid fixed point theorems involving only one or two operators with the mixed hypotheses from any two of the main three streams of algebra, topology and geometry. But to our best knowledge, there is no hybrid fixed point theorem involving three operators satisfying the hypotheses from the above main three streams of fixed point theory. The purpose of the present paper is to prove some general multi-valued hybrid fixed point theorems which combine multi-valued versions of the theorems of Banach, Schauder, and Tarski, and to discuss some application of these results to differential inclusions. Some results are also new even to the single-valued case of operator equations in Banach spaces. The rest of the paper is organized as follows. Section 2 deals with the preliminaries and some fundamental results needed in the sequel. Section 3 deals with the main hybrid fixed point theorems and some corollaries in the applicable form to differential and integral inclusions and equations. Section 4 discusses the existence results for perturbed initial value problems of first order differential inclusions and finally some concluding remarks are given in Section 5.